The Index of Isolated Umbilics on Surfaces of Non-Positive Curvature

The index of isolated umbilics on surfaces of non-positive curvature

Autor(en): Fontenele, Francisco / Xavier, Frederico

Objekttyp: Article

Zeitschrift: L'Enseignement Mathématique

Band (Jahr): 61 (2015)

Heft 1-2

PDF erstellt am: 11.10.2021

Persistenter Link: http://doi.org/10.5169/seals-630591

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http://www.e-periodica.ch LEnseignement Mathematique (2) 61 (2015), 139-149 DOI 10.4171/LEM/61-1/2-6

The index of isolated umbilics on surfaces of non-positive curvature

Francisco Fontenele and Frederico Xavier

Abstract. It is shown that if a C2 surface M c M3 has negative curvature on the complement of a point q e M, then the Z/2-valued Poincare-Hopf index at q of either distribution of principal directions on M — {q} is non-positive. Conversely, any non-positive half-integer arises in this fashion. The proof of the index estimate is based on geometric- topological arguments, an index theorem for symmetric tensors on Riemannian surfaces, and some aspects of the classical Poincare-Bendixson theory.

Mathematics Subject Classification (2010). Primary: 53A05, 53C12; Secondary: 37C10, 37E35.

Keywords. Index of an umbilical point, principal curvature lines, Caratheodory conjecture, Loewner conjecture.

To Professor Jorge Sotomayor, on the occasion of his 70th birthday

1. Introduction

The distributions of principal directions on a surface in I3, defined on the complement of the umbilical set (i.e., the locus where the principal curvatures coincide), have been the object of intense scrutiny since the early days of differential geometry. For both technical and geometric reasons, most of these investigations were conducted under the hypothesis that the surface is real analytic, or at least of class C3, although one needs only C2 regularity in order for the fields of principal directions to be continuous. The aim of this work is to establish an estimate for the local index of these fields, under a natural curvature restriction, but with optimal regularity: 140 F. Fontenele and F. Xavier

Theorem 1.1. Let McM3 be a C2 surface, q e M. Assume that the Gaussian curvature of M is negative at every point of M other than q. Then, the Z/2- valued Poincare-Hopf index at q of either distribution of principal directions on M — {q} is non-positive.

We observe that the theorem is sharp. Indeed, if the Gaussian curvature remains negative at q, then the distributions of principal directions extend continuously to M, and so the index is zero. On the other hand, given any negative number j e Z/2 one can construct a surface as in the statement of the theorem, even a minimal one (i.e., with vanishing mean curvature), that has an isolated umbilic of index j ([SX3]). When the surface in question is minimal, the conclusion in Theorem 1.1 can be verified using the holomorphic data in the Weierstrass representation of the surface. The main point of the present work is that, surprisingly, Theorem 1.1 applies to surfaces that are merely C2, and not only to those CM surfaces that are minimal. Since complex analysis is no longer available in this more general setting, new tools have to be introduced in order to estimate the index. Loosely speaking, the replacement for complex analysis is, when properly augmented, the classical qualitative theory of planar dynamical systems. Umbilic points are notoriously elusive geometric objects. For instance, on a surface of positive curvature the index of an isolated umbilic need not be positive. In other words, the "dual" statement of Theorem 1.1 does not hold. Indeed, inverting surfaces satisfying the hypotheses of Theorem 1.1 on suitable spheres - a process that does not change the index of the umbilic -, one can produce surfaces of positive Gaussian curvature exhibiting an umbilic whose index is any prescribed negative half-integer. By analogy with the above mentioned sharpness of Theorem 1.1, one might naively expect that every positive half-integer could be realized as the index of an isolated umbilic on a surface of positive curvature. In stark opposition to this expectation, it is actually predicted that on any sufficiently regular surface, without any curvature restrictions whatsoever, the index of an isolated umbilic should be at most one. This is the well-known local Caratheodory conjecture, also known as the Loewner conjecture. We refer the reader to [Bol], [Bro], [FNB]-[GH], [GS]-[Ham2], [Iva]-[Xav] for a sample of the many works, old and new, on this very challenging problem, as well as on various aspects of the global study of principal foliations. It is worth to point out that, if true, the local Caratheodory conjecture would imply, by the Poincare-Hopf theorem, the Caratheodory conjecture (the statement that any C2 immersion of the 2-sphere into M3 has at least two umbilical points; Isolated umbilics on surfaces of non-positive curvature 141 see [GK] for a solution of this conjecture that has been proposed recently). To put our results in perspective, we re-iterate that Theorem 1.1 is sharp and verifies the C2 version of the local Caratheodory conjecture in a geometrically important special case, but with a stronger conclusion. Thus, Theorem 1.1 represents a contribution to the interface between classical differential geometry and classical dynamical systems that stands on its own, since it cannot be subsumed by the resolution of the Caratheodory conjecture. On the other hand, we hasten to add that there is no expectation that the present method can be used to tackle the said conjecture, given our strong reliance on negative curvature. Over the years, the task of estimating the index of isolated umbilics has proven to be an arduous one, often involving lengthy and intricate arguments (e.g., [Iva]). Against this backdrop, it is pleasing that the proof of Theorem 1.1, albeit delicate in its own right, is rather conceptual. The arguments are based on an index theorem for abstract symmetric tensors on Riemannian surfaces, elements of the classical qualitative theory of two-dimensional dynamical systems, and a modicum of topology and classical differential geometry. Although the main result is new, the subject matter lends itself to a more expository style and, accordingly, full details are provided.

2. An index theorem for abstract symmetric tensors

A continuous symmetric tensor field A of type (1,1), defined on a Riemannian surface M, is said to have an isolated A-singularity at q e M if there exists a neighborhood V of q such that the eigenvalues of A(p) are unequal for any p e V — {q}. This condition is equivalent to the requirement that the traceless part A{p) — |(trA(p))I of A{p) be non-zero for all p ^ q. If the eigenvalues of A(q) are distinct, then q is automatically an isolated A-singularity (one then thinks of q as being a "removable" A-singularity). Under the above conditions, there are two continuous line fields on M — {q}, not necessarily orientable, which correspond to the diagonalizing directions of A. Given a continuous field of directions £ on V — {q}, denote by A£ the field of directions obtained by applying A{p) to any vector generating the one- dimensional subspace £(/?) of TPM. We write j(A) for the index at q of either of the two fields of diagonalizing directions of A. Similarly, j(rj) stands for the index at q of a line field rj with an isolated singularity at q. For the more general definition of local index of a totally real n-form (the case n 1 corresponds to a direction field), as well as the proof of the two-dimensional Poincare-Hopf theorem in this context, see [FNB]. 142 F. Fontenele and F. Xavier

Theorem 2.1. Let A be a continuous symmetric tensor field of type (7,7) on a Riemannian 2-manifold M, and q e M an isolated A -singularity. Then, for every continuous field of directions £ on a punctured neighborhood of q, one has

(2.1) 2j(A) j((A - IM)/) £) + _/(£)•

Remarks. One should think of £ as being a "test" line field. For instance, if £ is chosen to be one of the continuous fields of eigendirections of A, each term on the right hand side of (2.1) equals J (A). As another illustration to see that 2 is the correct factor in the left hand side, let A be a continuous symmetric tensor field on a compact orientable surface M of non-zero genus, with the property that the set F\ where A is a multiple of the identity is finite. Let £ be a continuous line field on M with a finite set F2 of singularities. Applying (2.1) around each point in Fi U F2, and summing, one sees from the Poincare-Hopf theorem that both sides of (2.1) equal 2/(M).

Proof Formula (2.1) was first established in [SX3], under the more restrictive assumptions that A is a smooth tensor and £ is a smooth vector field; here, A and £ are only assumed to be continuous, and £ is allowed to be an unorientable line field. This extra generality requires a new line of argument. Let X > /x be the eigenvalues of A and U a neighborhood of q such that X(p) > p(p) whenever p e U — {q}. The continuous distributions Dx and Dß of eigenspaces determined by A on U — {q} have the same index at q, and this common value is, by definition, the index j(A) of A at q. Let B A — ^(trA)7, so that trT? 0 everywhere, and let

a b b —a be the matrix representation of B with respect to an orthonormal frame {e\,e2} in (a possibly smaller neighborhod) U. Observe that the continuous vector field X := aei + be2 has no zeros on U — {q}. Let C c U be a Jordan curve around q and y : [0,1] U a positive parametrization of C (relative to the orientation determined by {e\,e2} on U). Let V(t) cos 0(0^1 + sin 9(t)e2 be a continuous unit vector field along y such that V(t) generates Dx(y(t)), for every t e [0,1]. By definition, 0(1) 0(0) (2.2) j(Dx) - 2tt (Notice that since L(l) ±L(0), 0(1) differs from 0(0) by a multiple of 7t, and thus j(Df) e ^Z.) Consider the continuous vector field W along y defined by Isolated umbilics on surfaces of non-positive curvature 143

(2.3) Wit) cos(2 0(0) ex + sin(2 0(0) e2.

We claim that W(t) is orthogonal to (—bei +a^2)(y(0)? f°r t e [0,1]. In fact, since V(t) is an eigenvector of A(y(t)) (and hence of B(y(t))) and sin 0(0

0 (B(V(t)), sin 0(0 e\ — cos 0(0 e2) (cos 6(t)(ae\ + be2) + sin 6(t)(be\ — ae2), sin 0(0 £i — cos 0(0 e2) — &(cos2 0(0 — sin2 0(0) + 2a sin 0(0 cos 0(0 —b cos(2 0(O) + sin(2 0(O) (cos (2 0(0) e\ + sin(2 0(0) £2, —+ 0^2) (2.4) (W(t),-bex +ae2), which proves the claim.

Since X is orthogonal to —be 1 +ae2, it follows from the claim above that

(2.5) 2L0y ±W. IA I Let £ be a continuous field of directions on U — {q}. If Z(t) cos (p(t)e\ + sin cp(t)e2 is a continuous vector field along y such that, for all t e [0,1], Z(t) generates £(y(0), then

(2-6) y(f) 2n From (2.3) and (2.5), we obtain

2?(Z(0)) [a cos

+ [sin(20) cos cp — cos(20) sin cp]e2) (2.7) ±|W|{ cos (2 0 -

(2.8) cos (2 9(t) — (p(t))e\ + sin (2 9(t) —

3. Gradients and degenerate local homeomorphisms

The lemma below is well known for the usual gradient of a planar function. Here, we work in the context of arbitrary Riemannian surfaces.

Lemma 3.1. Let f be a C1 function defined on an open set U of a C2 Riemannian surface M, and q e U an isolated critical point of f. Then, the Poincare-Hopf index of V/ at q is at most one. Furthermore, the index of V/ at q is one if and only if f has a strict local maximum, or minimum, at q. Proof Taking U to be a coordinate neighborhood, U

Under the hypotheses of Theorem 1.1, if q is an umbilic point then the Gaussian curvature necessarily has to vanish at q, and so the Gauss map is not a local dilfeomorphism. However, one can still prove that the Gauss map is open. More generally, using arguments from algebraic topology, it is possible to argue that a continuous map must be open if it is a local homeomorphism on the complement of a sufficiently "thin" subset of its domain ([Chu, p. 354]). Fortunately, in the special case that concerns us, an elementary proof is available:

Lemma 3.2. Let U cl" be open, n > 2, q e U, F : U ^M.n continuous. If the restriction of F to U — {q} is a local homeomorphism, then F is an open map.

Proof. (The simple example f(x) x2 shows the need to have n > 2.) Assume, by contradiction, that F is not an open map. Since the restriction of F to U — {q} Isolated umbilics on surfaces of non-positive curvature 145 is a local homeomorphism, there exists an open set U c U, with q e V, such that F(q) e dF(V). Let B be an open ball centered at q such that B c V. We are going to need claims (i) and (ii) below: (i) For every y £ dF(B), y ^ F{q), there exists x £ dB satisfying F(x) y. Indeed, let (x^) be a sequence in B with F(xk) -> y. Passing to a subsequence, we can suppose -> x e B. Hence F(xk) -> F(x), and so F{x) y. Since y ^ F{q), and the image of every point in B — {q} belongs to the interior of F(B), one has x £ dB. (ii) Every ball D centered at F{q) contains a point y e dF{B) distinct from F(g). Observe that F(q) £ dF(B), otherwise F(q) £ intF(B) C intF(V). Since F(q) £ D n dF(B), one cannot have D C F(B). Let then z e D — F(B). Using the continuity of F and the fact that the restriction of F to B — {q} is a local homeomorphism, we see that D flintF(2?) ^ 0. Since n > 2, one can choose w £ D n intF(B) such that F(q) lies outside the segment xvz joining w to z. Since Wz joins a point in the interior of F(B) to a point in the complement of F(B), it must contain a point y £ dF(B), which is necessarily distinct from F(q). Since y £ Wz c D, (ii) follows. Applying (ii) to a sequence of balls D centered at F(q), with radii tending to zero, one sees that there exists a sequence (y^) in dF(B), with yk ^ F{q) for all k, such that y^ F(q). By (i), y^ F(xk) for some sequence (xjc) in dB. Passing to a subsequence, we can assume that x^ x e dB. By continuity, F(xk) F(x), and so F{x) F(q). We have then found a point x £ dB c V — {q} whose image by F belongs to dF(V), contradicting the fact that the restriction of F to V — {q} is a local homeomorphism.

4. A special homotopy and the proof of Theorem 1.1

Let U C M be a neighborhood of q on which a C1 normal (Gauss) map £ : U S2 is defined. Choose a £ S2 such that 0 < {a,t=(q)) < 1. Shrinking J7, one may assume that 0 < (a,%(p)} <1 if p £U. Denote by a the second fundamental form of M, and consider the height function / : M R, f(x) {x,a). In particular, the (intrinsic) gradient and Hessian of / satisfy (4.1) V/O) a - (a,l(p))l(p), Hessf(p)(v,v) H(p),a)a(v,v), v e TPM.

The first equation expresses the gradient of the restriction as the orthogonal projection of the space gradient into the tangent space. The formula for the 146 F. Fontenele and F. Xavier

Hessian of the restriction of a function is standard in submanifold geometry, and can be found, say, in ([Daj, p. 46]). Writing Hf(p) and A(p) for the linear endomorphisms associated to the quadratic forms Hess/(/?) and a(p) on TPM, respectively, it is clear from (4.1) that the indices of the continuous symmetric tensors A and Hp satisfy j(Hf) j(A). Using the fact that det A, being the Gaussian curvature, is negative away from q by hypothesis, it is easy to see that

Mp)-^A(p))l, 0

Since t e [0,1] and the principal curvatures satisfy X(p) > 0 > fi(p) if p ^ q, it is clear that the diagonal elements are non-zero. It follows from the invariance of the degree under homotopies which do not introduce further zeros that, for every continuous non-vanishing vector field q on U-{q}9 j(Arj) j((A - i(trA)/)77). Hence, by (4.1), Theorem 2.1 and the fact that (%(p),a) ^ 0 for all p e U, (4.2) 2j{A) j(Arj) + j{n) j(Hfri) + j(r,).

Applying (4.2) with q V/ (which is a permissible choice, since 0 < {a,^(p)) < 1 implies V/(/>) ^ 0), and using the general formula

(4.3) v(i|V^|2), which is valid on any Riemannian manifold, with \j/ /, one has (4.4) 2j(A) j(HfVf) + j(Vf) j (v(i|V/|2)) + j(Vf).

Manifestly, (4.4) provides a formula for the index of the umbilic q (i.e., the index of the tensor field A) in terms of the indices of two gradient fields. To conclude the proof of the theorem, we must show that j(A) is non-positive. If q is not an umbilical (planar) point, the distributions of principal directions extend continuously across q, in which case j(A) vanishes. Hence, we may assume that q is an umbilic, that is, \{q) fi(q) 0. In particular, A(q) 0. Isolated umbilics on surfaces of non-positive curvature 147

Since V/(/7) ^ 0 for p e U, the term y(V/) in (4.4) vanishes. Hence, it remains to argue that y(V(^|V/12)) < 0. According to Lemma 3.1, one must show that the function h ^|V/|2 has an isolated critical point at q, which is neither a local maximum nor a local minimum. From (4.1) and (4.3), one has Vh(p) V(^|Vf\2)(p) (a,^(p))A(p)Vf(p). Since A(q) 0, the point q is critical for h. In order to see that no point p in U — {q} is critical for A, observe that {a,t=(p)} > 0, V/(/?) ^ 0 and A(p) is invertible (since the Gaussian curvature det A(p) is negative for p ^ q, by hypothesis). Hence q is an isolated critical point of A. We now proceed to show that A has neither a local maximum nor a minimum at q. A direct calculation gives y/2h(p) |V/(/?)| sin 6(p), where 6{p) e [0, jv] is the angle between the vectors a and i-(p). Since 0 < (a,%(q)) < 1, it follows that 6{q) e (0, j). Therefore, in order to show that h does not have an extremum at q, it suffices to argue that the Gauss map £ U ^ S2 is open. But this is a consequence of Lemma 3.2 and the inverse function theorem, since the Jacobian determinant of the normal map is the Gaussian curvature which, by hypothesis, is negative away from q.

Acknowledgements. The work of Fontenele was partially supported by CNPq (Brazil). The work of Xavier was partially supported by IMPA and CAPES (Brazil).

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(Regu le 18 mars 2014)

Francisco Fontenele, Departamento de Geometria, Universidade Federal Fluminense, Niteröi, RJ 24020-140, Brazil e-mail: [email protected] Isolated umbilics on surfaces of non-positive curvature

Frederico Xavier, Department of Mathematics, Texas Christian University, Fort Worth, TX 76129, USA e-mail: [email protected]